An optimal lower bound for monotonicity testing over hypergrids

TL;DR

This paper establishes a tight lower bound on the number of queries needed for adaptive monotonicity testing of functions over hypergrids, matching known upper bounds and resolving a key open problem.

Contribution

It proves an optimal lower bound of a9^{-1}d \u2212 a9^{-1} queries for adaptive monotonicity testing over hypergrids, closing the gap with existing upper bounds.

Findings

Lower bound matches existing upper bounds.

Establishes optimal query complexity for hypergrid monotonicity testing.

Resolves an open problem in property testing for hypergrids.

Abstract

For positive integers $n, d$, consider the hypergrid $[n]^d$ with the coordinate-wise product partial ordering denoted by $\prec$. A function $f: [n]^d \mapsto \mathbb{N}$ is monotone if $\forall x \prec y$, $f(x) \leq f(y)$. A function $f$ is $\eps$-far from monotone if at least an $\eps$-fraction of values must be changed to make $f$ monotone. Given a parameter $\eps$, a \emph{monotonicity tester} must distinguish with high probability a monotone function from one that is $\eps$-far. We prove that any (adaptive, two-sided) monotonicity tester for functions $f:[n]^d \mapsto \mathbb{N}$ must make $\Omega(\eps^{-1}d\log n - \eps^{-1}\log \eps^{-1})$ queries. Recent upper bounds show the existence of $O(\eps^{-1}d \log n)$ query monotonicity testers for hypergrids. This closes the question of monotonicity testing for hypergrids over arbitrary ranges. The previous best lower bound for general hypergrids was a non-adaptive bound of $\Omega(d \log n)$.